Theory of a 3+1D fractional chiral metal: Interacting variant of the Weyl semimetal

Formulating consistent theories describing strongly correlated metallic topological phases is an outstanding problem in condensed-matter physics. In this work, we derive a theory defining a fractionalized analog of the Weyl semimetal state: the fractional chiral metal. Our approach is to construct a 4+1D quantum Hall insulator by stacking 3+1D Weyl semimetals in a magnetic field. In a strong enough field, the low-energy physics is determined by the lowest Landau level of each Weyl semimetal, which is highly degenerate and chiral, motivating us to use a coupled-wire approach. The one-dimensional dispersion of the lowest Landau level allows us to model the system as a set of degenerate 1+1D quantum wires that can be bosonized in the presence of electron-electron interactions and coupled such that a gapped phase is obtained whose response to an electromagnetic field is given in terms of a Chern-Simons field theory. At the boundary of this phase, we obtain the field theory of a 3+1D gapless fractional chiral state, which we show is consistent with a previous theory for the surface of a 4+1D Chern-Simons theory. The boundary's response to an external electromagnetic field is determined by a chiral anomaly with a fractionalized coefficient. We suggest that such an anomalous response can be taken as a working definition of a fractionalized strongly correlated analog of the Weyl semimetal state.

Figure 1

Construction of a 4+1D integer quantum Hall effect from coupled Weyl semimetals. A tunnel coupling between the left-handed node of each Weyl semimetal with the right-handed node of its neighbor at smaller $x_4$, depicted by red arrows, induces a gap for all bulk nodes. In a slab of finite extent $0\le x_4\le L_4$, single gapless nodes remain at the 3+1D surfaces at $x_4=0$ and $x_4=L_4$.

Figure 2

Stacking of Weyl semimetals along the fourth spatial dimension $x_4$ subject to a magnetic field $\mathbf{B}=B_3{\mathbf{e}}_3$. Each Weyl semimetal contains the Landau levels of a pair of Weyl nodes. The left panels depicts the energy $E_0^{\chi }$ of a Weyl node of chirality $\chi =\pm$ in a magnetic field described by the Hamiltonian $H_0^{\chi }$ as a function of the momentum $p_3$ measured with respect to the Weyl node. The Landau levels $n\ge 1$ form quadratically dispersing, particle-hole symmetric bands. Each Landau level is macroscopically degenerate with respect to $p_2$. The right panel illustrates the low-energy description of the stacked 4+1D system at finite $\mathbf{B}$, which reduces to the gapless lowest Landau levels. As discussed in Sec. 3b, a complex hopping $t{e}^{i\mathrm{\Delta }{p}_3{x}_3}$ connects neighboring Weyl semimetals.

Figure 3

Correlated tunnelings between neighboring Weyl semimetals leading to fractional quantum Hall states. While an electron hops from the left-handed Weyl node at $x_4=(q+1)a_4$ to the right-handed node at $q{a}_4, m$ electrons are scattered from the left-handed node to the right-handed node in both Weyl semimetals connected by the tunneling. This correlated process is indicated by the arrows, whose labels indicate the number of electrons transported along the respective arrow.

Figure 4

Generation of the correlated scattering $\sim t{U}^2$ corresponding to $m=1$ in Fig. 3. Empty circles depict the initial states of the electrons, filled circles correspond to their final states, and the dotted circles indicate the intermediate virtual states of the hopping electron. The dotted horizontal lines depict the Fermi level (and thus the energy of the Weyl nodes).

Figure 5

Two of the possible competing interaction processes that would lead to a charge-density-wave order (top panel), and a 1/3-Laughlin-crystal-type order (bottom panel) [23].

Figure 6

(a) Upper panel: wire construction of the 2+1D quantum Hall effect. Lower panel: each wire has a quadratic dispersion relation that is gapped by interwire tunneling terms at integer filling, depicted by the dashed line. The lowest band is composed of boundary left and right movers denoted by blue and red filled circles, respectively, distinct from bulk states (black circles). (b) When an electric field is applied, left movers are pumped to right movers through the bulk states. The nonconservation of left and right chiralities, manifested through the distinct left and right chemical potential depicted as dashed lines, is the $1+1\mathrm{D}$ chiral anomaly. Our construction can be thought of as $N_{\text{LL}}$ copies of this state, each contributing with its own chiral anomaly. The surface states of the 4+1D quantum Hall effect constructed in this way define the chiral metal at the boundary (see the main text).